An edge colouring of a graph G is complete if for any distinct colours c1\documentclass[12pt]{minimal}
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\begin{document}$$c_1$$\end{document} and c2\documentclass[12pt]{minimal}
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\begin{document}$$c_2$$\end{document} one can find in G adjacent edges coloured with c1\documentclass[12pt]{minimal}
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\begin{document}$$c_1$$\end{document} and c2\documentclass[12pt]{minimal}
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\begin{document}$$c_2$$\end{document}, respectively. The pseudoachromatic index of G is the maximum number of colours in a complete edge colouring of G. Let ψ(n)\documentclass[12pt]{minimal}
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\begin{document}$$\psi (n)$$\end{document} denote the pseudoachromatic index of Kn\documentclass[12pt]{minimal}
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\begin{document}$$K_n$$\end{document}. In the paper we proved that if x≥2\documentclass[12pt]{minimal}
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\begin{document}$$ x\ge 2 $$\end{document} is an integer and n∈{4x2-x,⋯,4x2+3x-3}\documentclass[12pt]{minimal}
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\begin{document}$$n\in \{4x^2-x,\dots ,4x^2+3x-3\}$$\end{document}, then ψ(n)≤2x(n-x-1)\documentclass[12pt]{minimal}
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\begin{document}$$\psi (n) \le 2x(n-x-1)$$\end{document}. Let q be an even integer and let ma=(q+1)2-a\documentclass[12pt]{minimal}
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\begin{document}$$ m_a=(q+1)^2-a $$\end{document}. If there is a projective plane of order q, a complete edge colouring of Kma\documentclass[12pt]{minimal}
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\begin{document}$$K_{m_a}$$\end{document} with (ma-a)q\documentclass[12pt]{minimal}
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\begin{document}$$(m_a-a)q$$\end{document} colours, a∈{-1,0,⋯,q2+1}\documentclass[12pt]{minimal}
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\begin{document}$$ a\in \{-1,0,\dots ,\frac{q}{2}+1\}$$\end{document}, is presented. The main result states that if q≥4\documentclass[12pt]{minimal}
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\begin{document}$$q\ge 4$$\end{document} is an integer power of 2, then ψ(ma)=(ma-a)q\documentclass[12pt]{minimal}
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\begin{document}$$\psi (m_a)=(m_a-a)q$$\end{document} for any a∈{-1,0,⋯,1+4q+92-1}.\documentclass[12pt]{minimal}
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\begin{document}$$ a\in \{-1,0,\dots ,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} .$$\end{document}
